From Quadratic Reciprocity to the Langlands Program

From quadratic reciprocity to the Langlands program

(Oliver Lorscheid, April 2018)

In honor of the 2018 Abel prize winner Robert Langlands What is the Langlands program?

The Langlands program consists of a web of conjectures, partially relying on conjectural objects, that connect L-functions for Galois representations and automorphic representations.

A simple, but yet unproven and tantalizing instance is the following. Conjecture (Langlands 1970) For every positive integer n, there is a bijection  irreducible automorphic  irreducible representations   Φ cuspidal representations GL −→ ρ : LQ → n(C)  π of GLn over Q  such that if where is the L(ρ, s) = L(π, s) π = Φ(ρ) LQ (conjectural) Langlands group of Q, which is an extension of absolute Galois group Gal(Q/Q) of Q. Part 1: Reciprocity laws and class eld theory Quadratic reciprocity

Let p be an odd prime number and a an integer that is not divisible by p. The Legendre symbol of a mod p is ( a 1 if a is a square modulo p, = p −1 if not.

Conjecture (Euler 1783, Legendre 1785) Let and be distinct odd primes. Then p q Adrien-Marie Legendre

q p p−1 q−1 = (−1) 2 2 . p q

I First complete proof by Gauÿ in 1796.

I Up to today, there are more than 240 dierent proofs. Carl Friedrich Gauÿ The p-adic numbers

Let p be a prime number. The p-adic absolute value i a −i 0 is dened as whenever is not |·|p : Q → R≥ p b p = p ab divisible by p.

The p-adic numbers are the completion Qp of Q with respect to the norm . |·|p The -adic integers are 1 . p Zp = a ∈ Qp |a|p ≤

Since R is the completion of Q at the archimedean absolute value sign , we often write and to |a|∞ = (a) · a Q∞ = R p ≤ ∞ express that p is a prime number or the symbol ∞. Hilbert reciprocity Let a and b be rational numbers and p ≤ ∞. The Hilbert symbol is

(1 if 2 2 2 has a solution 3 ax + by = z (x, y, z) ∈ Qp, (a, b)p = −1 if not.

Theorem (Hilbert 1897)

For all a, b ∈ Q, we have (a, b)p = 1 for almost all p and Y (a, b)p = 1. p≤∞ David Hilbert Remark: Note that this implies quadratic reciprocity since for distinct odd prime numbers p and q, we have

Y p−1 q−1 q p (p, q) = (p, q)2(p, q) (p, q) = (−1) 2 2 . ` p q p q `≤∞ Class eld theory and Artin reciprocity

Theorem (Takagi 1920, Artin 1927)

Let L/Qp be a nite Galois extension with abelian Galois group G = Gal(L/Qp) and norm N : L → Qp, dened by Q . Then there is a N(a) = σ∈G σ(a) group isomorphism Teiji Takagi

× × ∼ Gal (−, L/Qp): Qp /N(L ) −→ (L/Qp), which is called the local Artin symbol. Theorem (Artin 1927) Let L/Q be a Galois extension with abelian Galois Emil Artin group Gal ). Then Q 1 G = (L/Q p≤∞(a, L Qp/Qp) = × for all a ∈ Q . Remark: This generalizes Hilbert's reciprocity law since √ (a,L / )( b) √ (a, b) = Qp√Qp where L = [ b]. p b Q Chevalley's idelic formulation

The idele group of Q is the group Q × × for almost all IQ = (ap) ∈ p≤∞ Qp ap ∈ Zp p < ∞ . Note that × embeds diagonally into . Q IQ The idele class group of is ×. Q CQ = IQ/Q

Theorem (Chevalley 1936) Claude Chevalley ab Let Q be the maximal abelian extension of Q. The product over all local Artin symbols denes a surjective group homomorphism

Gal ab r : CQ −→ (Q /Q) whose kernel is the connected component C 0 of the identity Q component of (w.r.t. the idelic topology on ). CQ CQ Part 2: L-series The Euler product

In 1740, Euler describes explicit formulas for

X 1 ζ(s) = ns n≥1

where s is an even positive integer. The case s = 2 solved the long standing Basel problem. Leonard Euler Moreover, Euler showed that

Y 1 ζ(s) = 1 − p−s p<∞

for real numbers s > 1, which is known as the Euler product nowadays. Dirichlet series

× Let χ : Z → C be a group homomorphism of nite order, i.e. χ(n) = 1 for some n ≥ 1. The Dirichlet series of χ is

X χ(n) Y 1 L(χ, s) = = . ns 1 − χ(p)p−s n≥1 p<∞

Application to primes in arithmetic progression:

Peter Gustav Lejeune Theorem (Dirichlet 1840) Dirichlet Let a and n be positive integers. Then there are innitely many prime numbers p congruent to a modulo n. Riemann's analysis of ζ(s)

Theorem (Riemann 1856) As a complex function, ζ(s) converges absolutely in the halfplane {s ∈ C|Re(s) > 1}, has a meromorphic continuation to C with a simple pole at 1, and it satises a functional equation of the form

Bernard Riemann ζ(1 − s) = (well-behaved factor) · ζ(s).

In the following, we shall refer to such properties as arithmetic.

In honor of Riemann's 1856 paper, ζ(s) is called the Riemann zeta function nowadays. Not to forget the Riemann hypothesis: Conjecture (Riemann 1856) If ζ(s) = 0, then s is an even negative integer or Re(s) = 1/2. Hecke L-functions Generalization of Dirichlet series (here only for Q): A Hecke character is a continuous group homomorphism ×. It is unramied at χ : CQ → C p if the composition × × is trivial. Zp → CK → C The Hecke L-function of χ is

Y 1 L(χ, s) = . Erich Hecke 1 − χ(p)p−s χ unramied at p

Theorem (Hecke 1916) L(χ, s) is arithmetic. If χ is nontrivial, then L(χ, s) is entire, i.e. without pole at 1.

Problem: How to factorize large L-functions into smaller L-functions? Artin L-functions

Let K/Q be a Galois extension with Galois group G = Gal(K/Q). An n-dimensional Galois representation of G is a continuous group homomorphism

ρ : G −→ GLn(C).

If n = 1, then ρ is called a character.

The Artin L-function of ρ is dened as the product Y L(ρ, s) = Lp(ρ, s) p<∞

of local factors Lp(ρ, s), which are, roughly speaking, the reciprocals of the characteristic polynomials of ρ(Frobp) where Frobp is a lift of the Frobenius automorphism of the residue eld. Artin reciprocity, part 2

As a consequence of Artin's and Chevalley's theorems, the reciprocity map Gal ab induces an isomorphism r : CQ → (Q /Q) ∗ Hom Gal ab × ∼ Hom 0 × r : ( (Q /Q), C ) −→ (CQ/CQ, C ) between the respective character groups via r ∗(ρ) = ρ ◦ r. Theorem (Artin 1927) If χ = ρ ◦ r, then L(χ, s) = L(ρ, s).

Applications: (1) Artin L-functions are arithmetic. (2) Factorization of large L-functions into smaller L-functions. The abelian Langlands correspondence

Let Gal be the absolute Galois group of . GQ = (Q, Q) Q Since Gal( ab/ ) = G ab and since every group homomorphism Q Q Q G → × factors through G ab, we gain an isomorphism Q C Q Hom × Hom Gal ab × ∼ Hom 0 × (GQ, C ) = ( (Q /Q), C ) −→ (CQ/CQ, C ). To extend this to an isomorphism with the whole character group of , we have to exchange by the Weil group of . CQ GQ WQ Q

Theorem (Langlands correspondence for GL1) Artin reciprocity induces an isomorphism

Hom × ∼ Hom × Φ: (WQ, C ) −→ (CQ, C ) such that if . L(ρ, s) = L(χ, s) χ = Φ(ρ) André Weil Part 3: Going nonabelianthe Taniyama-Shimura-Weil conjecture Dirichlet series of a modular form

The Poincaré upper half plane is H = {z ∈ C|Im(z) > 0}. A modular cusp form (of weight k and level N) is an holomorphic function f : H → C of the form Henri Poincaré ∞ X 2 az + b f (z) = a e πinz/N s.t. f = (cz + d)k f (z) n cz + d n=1 for all a b SL congruent to 1 0 modulo . c d ∈ 2(Z) 0 1 N

It is a (Hecke) eigenform if apq = apaq for all primes p 6= q. The Dirichlet series of f is X an Y 1 L(f , s) = s = −s . n 1 − app n≥1 p<∞ Theorem (Hecke 1936) L(f , s) is arithmetic. Elliptic curves

1 1 An elliptic curve over Q is a Lie group E ' S × S (i.e. a complex torus) that is dened by equations over Q. In particular, it contains a subgroup on which acts. E(Q) ⊂ E GQ n n For a prime ` and n ≥ 1, let E(Q)[` ] be the subgroup of ` -torsion points of E(Q). The Tate module of E is

T (E) = lim E( )[`n], ` ←− Q n which is isomorphic to 2 and has an action of . Z` GQ For every good prime ` and embedding Z` → C, we get a Galois representation John Tate

GL GL2 ρ : GQ −→ (T`(E) ⊗Z` C) ' (C). The L-function of E is the Artin L-function L(E, s) = L(ρ, s). The Taniyama-Shimura-Weil conjecture

Conjecture (Taniyama 1955, Shimura 1957, Weil 1967) For every elliptic curve E over Q, there is a modular cusp eigenform f of weight 2 such that L(E, s) = L(f , s). The conjecture was proven, step-by-step, by

I Weil (1967),

I Wiles (1995), with some help of Taylor,

I Diamond (1996), Yukata Taniyama

I Conrad, Diamond, Taylor (1999),

I Brevil, Conrad, Diamond, Taylor (2001). Nowadays it is called the modularity theorem.

Remark: Wiles contribution gained popularity since

it implied Fermat's last theorem. Wiles received Goro Shimura the Abel prize in 2016. The modularity theorem as a Langlands correspondence

Since every modular cusp eigenform f (of some level N) is an element of an irreducible automorphic cuspidal representation π and L(π, s) = L(f , s), the modularity theorem can be rephrased as follows.

There exists a bijection

 certain   certain       irreducible automorphic  irreducible representations Φ −→ cuspidal representations  ρ : G → GL2(C)    Q  π of GL2 over Q  such that L(ρ, s) = L(π, s) if π = Φ(ρ). Part 4: The Langlands program Adele groups

The adele ring of Q is Y A = (ap) ∈ Qp ap ∈ Zp for almost all p < ∞ . p≤∞

Note that × GL . In general, IQ = A = 1(A) Y GLn(A) = (gp) ∈ GLn(Qp) gp ∈ GLn(Zp) for a.a. p < ∞ p≤∞ is equipped with a topology, for which Y K = On × GLn(Zp) p<∞

is a maximal compact subgroup. In particular, GLn(A) is a locally compact group and carries a Haar measure, which allows us to form integrals over subgroups. Automorphic cusp forms

An (automorphic) cusp form for GLn over Q is a smooth function f : GLn(A) −→ C such that 0 I there is a nite index subgroup K of K such that 0 f (γgk) = f (g) for all γ ∈ GLn(Q) and all k ∈ K ; I the constant Fourier coecient Z c0(f )(g) = f (ug) du

U(A)

vanishes for every g ∈ GLn(A) and every unipotent subgroup U of GLn. Joseph Fourier Analogy: A modular form P∞ 2πinz/N is a cusp form if f = n=0 an e and only if a0 = 0. Automorphic cuspidal representation

The space A0 of all cusp forms for GLn over Q is almost a representation of GLn(A) with respect to the action g.f (h) = f (hg) where g, h ∈ GLn(A) and f ∈ A0.

An automorphic cuspidal representation of GLn over Q is a subrepresentation π of A0.

Note that A0 decomposes into a direct sum of irreducible subrepresentations π. Example: Every modular cusp eigenform generates an irreducible automorphic cuspidal representation π for GL2 over Q. L-functions Langlands denes an L-function L(π, s) for every irreducible automorphic cuspidal representation π in terms of the Satake parameters of π. Conjecture (Langlands 1970) L(π, s) is arithmetic.

coecients ap for p prime. If π is the corresponding automorphic representation, then

Y 1 L(π, s) = L(f , s) = . 1 − a p−s p<∞ p A rst Langlands conjecture

Conjecture (Langlands 1970) For every positive integer n, there is an injection  irreducible automorphic  irreducible representations   Φ cuspidal representations GL −→ ρ : GQ → n(C)  π of GLn over Q  such that if where Gal is L(ρ, s) = L(π, s) π = Φ(ρ) GQ = (Q/Q) the absolute Galois group of Q.

In order to recover the missing Galois representation, Langlands suggests the existence of a certain extension of , coined as the LQ GQ Langlands group nowadays. A rst Langlands conjecture

Conjecture (Langlands 1970) For every positive integer n, there is a bijection  irreducible automorphic  irreducible representations   Φ cuspidal representations GL −→ ρ : LQ → n(C)  π of GLn over Q  such that if where is the L(ρ, s) = L(π, s) π = Φ(ρ) LQ (conjectural) Langlands group of Q.

In order to recover the missing Galois representation, Langlands suggests the existence of a certain extension of , coined the LQ GQ Langlands group nowadays. Automorphic L-functions In fact, Langlands proposes a common generalization of Artin L-functions and L-functions of automorphic representations for any reductive group G over any local or global eld. To give an idea: the Satake parameters of an automorphic representation π of GLn(A) are the entries of certain diagonal matrices Aπ,p in GLn(C) (one for each p < ∞). L The Langlands dual G of G = GLn(A) is an extension of Gal(Q/Q) by GLn(C):

L 1 −→ GLn(C) −→ G −→ Gal(Q/Q) −→ 1.

The automorphic L-function L(π, ρ, s) is dened for every irreducible automorphic cuspidal representation π of G and every L representation ρ : G → GLN (C). Conjecture (Langlands 1970) L(π, ρ, s) is arithmetic. Langlands functoriality

Let k be a local or global eld (e.g. a p-adic eld or a number 0 eld) and G and G reductive groups over k (e.g. GLn, SOn or Sp ) with respective Langlands duals L and L 0. 2n G G An L-morphism is a continuous group homomorphism L L 0 ϕ : G → G that commutes with the respective projections to Gk . Conjecture (Langlands 1970) Let ϕ : LG → LG 0 be an L-morphism and π an irreducible automorphic cuspidal representation of G. Then 1. there is an irreducible automorphic cuspidal representation π0 of G 0 whose Satake parameters are the images of the Satake parameters of π, and 0 L 0 0 2. for every representation ρ : G → GLN (C) and ρ = ρ ◦ ϕ, we have L(π0, ρ0, s) = L(π, ρ, s). Part 5: What has been done? 1970 Jaquet, Langlands: correspondence for GL2 over p-adic elds (p 6= 2) 1973 Langlands: correpondence for GL2 over R and C 1977 Drinfeld: global correspondence for GL2 over global function elds (awarded with a Fields medal)

1980 Kutzko: correspondence for GL2 over all local elds

1993 Laumon, Rapoport, Stuhler: correspondence for GLn over local elds of positive characteristic 1995 Wiles: modularity theorem (awarded with the Abel prize)

2000 L. Laorgue: global correspondence for GLn over global function elds (awarded with a Fields medal)

2001 Harris, Taylor: correspondence for GLn over local elds of characteristic 0

2000 Henniart: alternative proof for GLn over local elds of characteristic 0 2008 Ngô: fundamental lemma (awarded with a Fields medal)

2013 Scholze: alternative proof for GLn over local elds of characteristic 0